1. Posterior Updating (Or: Is my cat a picky eater?)
2. Priors for estimation
3. Priors for model comparison (Or: Is my cat fat?)
4. Prior prediction and prior sensitivity

• This is Frank.
• Frank likes to eat but he might be a tad picky.
• We may model how often Frank eats his food in a month: \[Y \sim \mbox{Binomial}(\theta, 30)\]

Is Frank a picky eater?

\[Y \sim \mbox{Binomial}(\theta, 30),\] \[\theta = .5.\]

\[Y \sim \mbox{Binomial}(\theta,30), \qquad \theta = .5.\]

Let’s say he ate 21 out of 30 meals. \(Y = 21.\)

In Bayesian statistical analysis we typically would use a prior distribution for parameters. \[ \begin{aligned} Y|\theta &\sim \mbox{Binomial}(\theta,N),\\ \theta &\sim \mbox{Beta}(a,b). \end{aligned} \]

If we assume Frank will most likely eat 5 out of 10 meals we may use \(a = 5\) and \(b = 5\).

from \(Pr(\theta)\)…

from \(Pr(\theta)\)… to \(Pr(\theta|Y)\).

\[ Pr(\theta|Y) = Pr(\theta)\frac{Pr(Y|\theta)}{Pr(Y)} \]

• \(Pr(\theta|Y)\) is the posterior distribution of \(\theta\).
• \(Pr(\theta)\) is the prior distribution of \(\theta\).
• \(Pr(Y|\theta)\) is the likelihood of the data.
• \(Pr(Y)\) is the marginal likelihood.

What should it look like?

• “Priors should represent your belief about the parameter.”
• “Priors should represent what experts in the field might expect.”
• “Priors should contain as little information as possible. Let the data speak.”
• “Priors should represent reasonable expectation about likely ranges of the parameter.”

• There are priors that are most suitable for estimation.
• And there are priors most suitable for model comparison.
• And there are priors that are pretty good for both.
• Oh, and not everyone agrees on this classifications (or what “good means”).

• Flat priors.
• Jeffreys priors.
• Reference priors.
• (Default priors.)

\[\theta \sim \mbox{Uniform}(0, 1).\]

theta <- runif(100000)
hist(theta, probability = T)

What are the odds of Frank eating the food, \(OR = \frac{\theta}{1 - \theta}\)?

Suppose \(\theta\) does not have constrained support (mean difference between two measures): \[\theta \sim \mbox{Uniform}(-\infty, \infty).\]

theta <- runif(100000, min = -100000, max = 100000)
hist(theta, probability = T, ylim = c(0, 1))

• Goal: Allow the data to have the maximum effect on the posterior estimates.
• Process: Maximize divergence from hypothetical data.
• The same for one-dimensional parameters.

Frank eats his food 7 out of 10 times.

Frank eats his food 21 out of 30 times.

Frank eats his food 650 out of 1000 times.

Frank might be picky, but once we figured out what he likes he is getting pretty fat. So we put him on a diet for 30 days.

• \({\cal M}_1\): Frank loses weight because of the diet.
• \({\cal M}_0\): Frank does not lose weight because of the diet.
• Change in weight might be normally distributed: \(Y_i \sim \mbox{Normal}(\mu, \sigma^2)\)
• For model \({\cal M}_1\) we might want a truncated normal prior on mu: \(\mu \sim \mbox{Normal}_{-}(a, b^2)\)
• For model \({\cal M}_0\) \(\mu = 0\).
• Prior on \(\sigma^2\)?
• Prior settings on \(a\) and \(b\)?

• \({\cal M}_1\): Frank loses weight because of the diet.
• \({\cal M}_0\): Frank does not lose weight because of the diet.
• Change in weight might be normally distributed: \(Y_i \sim \mbox{Normal}(\mu, \sigma^2)\)
• For model \({\cal M}_1\) we might want a truncated normal prior on mu: \(\mu \sim \mbox{Normal}_{-}(a, b^2)\)
• For model \({\cal M}_0\) \(\mu = 0\).
• Prior on \(\sigma^2\)?
• Prior settings on \(a\) and \(b\)?

1/30

## [1] 0.03333333

• \({\cal M}_1\): Frank loses weight because of the diet.
• \({\cal M}_0\): Frank does not lose weight because of the diet.
• Change in weight might be normally distributed: \(Y_i \sim \mbox{Normal}(\mu, \sigma^2)\)
• For model \({\cal M}_1\) we might want a truncated normal prior on mu: \(\mu \sim \mbox{Normal}_{-}(a, b^2)\)
• For model \({\cal M}_0\) \(\mu = 0\).
• Prior on \(\sigma^2\)?
• Prior settings on \(a\) and \(b\)?

• BayesFactor package in R and JASP use default priors for the t-test called JZS prior.
• Prior structure is constant.
• Prior setting is on the scale of the effect size (think Cohen’s \(d\)).

• BayesFactor package in R and JASP use default priors for the t-test called JZS prior.
• Prior structure is constant.
• Prior setting is on the scale of the effect size (think Cohen’s \(d\)).

BayesFactor::ttestBF(x = Change[-1]
                     , mu = 0
                     , nullInterval = c(-Inf, 0)
                     , rscale = 1/sqrt(2))

BayesFactor::ttestBF(x = Change[-1]
                     , mu = 0
                     , nullInterval = c(-Inf, 0)
                     , rscale = 1/sqrt(2))

BayesFactor::ttestBF(x = Change[-1]
                     , mu = 0
                     , nullInterval = c(-Inf, 0)
                     , rscale = 1/3)

BayesFactor::ttestBF(x = Change[-1]
                     , mu = 0
                     , nullInterval = c(-Inf, 0)
                     , rscale = 1/3)

## Bayes factor analysis
## --------------
## [1] Alt., r=0.333 -Inf<d<0    : 3.095844  ±0%
## [2] Alt., r=0.333 !(-Inf<d<0) : 0.1448153 ±0%
## 
## Against denominator:
##   Null, mu = 0 
## ---
## Bayes factor type: BFoneSample, JZS

• How does your prior influence the results of the analysis?
• Redo the analysis for a range of priors
• If the results are (relatively) stable then we may trust them more.

• How does your prior influence the results of the analysis?
• Redo the analysis for a reasonable range of priors.
• If the results are (relatively) stable then we may trust them more.

• What data are predicted by your prior?
• Are these predictions plausible?

\[Y \sim \mbox{Binomial}(\theta,30), \qquad \theta = .5.\]

• Example: Memory performance in a two-alternative forces choice recognition task (K = 100).
• Proposed prior for the probability of a correct response: \(\theta \sim \mbox{Beta}(2,2)\).
• Procedure: Sample from the prior, simulate data based on the sample.

• Example: Memory performance in a two-alternative forces choice recognition task (K = 100).
• Proposed Prior: \(\theta \sim \mbox{Beta}(2,2)\)
• Procedure: Sample from the prior, simulate data based on the sample.

M <- 1000 # simulation runs
p.theta <- rbeta(M, 2, 2) # theta sampled from the prior
y <- rbinom(1000, 100, p.theta)
hist(y)

• Using very wide normal priors (e.g., \(\mbox{Normal}(0, 100)\) for likert-scale data).
• When priors are placed on transformed parameters (e.g., logistic regression models).